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Index

Indexalgebra

Calkin, 4, 55complex, 1factor, 2quotient, 2real, 1with involution, 1

algebra Ainvolution on, 6

algebra T , 172algebra Tν , 184Allan’s local principle, 44approximate solution

biharmonic problem, 262

Banach algebracomplex, 2real, 2strictly real, 44

basesinterpolation, 88

biharmonic Dirichlet problem, 237boundary problem

Riemann-Hilbert, 123boundary value problem

exterior, 252for two analytic functions, 234

Calkin algebra, 55Carleman shift, 10center, 44, 81C∗-algebra

complex, 2real, 2

strictly real, 44, 58circulant, 74circulants

paired, 76collocation equations, 87collocation method

polynomial, 65collocation operator, 87commutant, 41complex spectrum, 16

description, 19complexification, 16condition

separation, 30constant

Poisson, 242coset, 2Costabel’s decomposition, 51cut-off method, 199

de Boor inequalities, 183decomposition

Costabel, 51double layer potential

equation, 202operator, 202

elementnormal, 3self-adjoint, 3

M -equivalentfrom the left, 41from the right, 41

M -invertiblefrom the left, 41

304 Index

ε-collocation method, 86equation

biharmonic, 235Cauchy singular integral, 59collocation, 87double layer potential, 202Muskhelishvili, 233singular integro-differential, 69

essential spectrum, 4exact sequence, 118extension Am, 6

factor algebra, 2factor para-algebra, 39Fourier transform, 48Fredholm operator, 3function

Holder continuous, 235upper semi-continuous, 42

fundamental boundary problemfirst, 241second, 242

Galerkin method, 123polynomial, 60

generalized Riemann-Hilbert-Poincare problem, 69

generalized Hankel operator, 50Gohberg-Krupnik local principle, 41Gohberg-Krupnik symbol, 84Goursat representation, 233

Holder space, 235homomorphism, 2

complex algebra, 2∗-algebra, 2

ideal, 2Np(α), 51

index, 3, 171initial projection, 30interpolation bases, 88interpolation method, 131interpolation projection, 87

inverse, 2left, 2Moore-Penrose, 21right, 2

inverse closed, 17invertibility

Moore-Penrose, 20, 86weak asymptotic Moore-

Penrose, 33involution, 1

on Ladd(H), 12on Ladd(L2(Γ0)), 9m-involution, 6

Laplace operator, 152local principle, 41

Allan, 44for para-algebras, 45Gohberg-Krupnik, 42

local symbol, 83localizing class, 41

of the para-algebraleft-, 45right-, 46

Lyapunov curve, 53piecewise, 53

m-closed subalgebra, 19m-involution, 6Mellin convolution operator, 49Mellin operator, 49

symbol, 50Mellin transform, 49mesh

graded, 181method

cut-off, 199ε-collocation, 86Galerkin, 123interpolation, 131polynomial qualocation, 100qualocation, 170spline Galerkin , 91spline qualocation, 104

Index 305

methodsquadrature, 93

Moore-Penroseinverse, 21invertibility, 20

Moore-Penrose invertibility, 86Muskhelishvili equation, 233

collocation method, 259, 275Galerkin method, 257, 265, 270,

275qualocation method, 261

Muskhelishvili operator, 244

normal element, 3

operatoradditive, ix, 9antilinear, 9asymptotically Moore-Penrose

invertible, 32block Mellin, 246collocation, 87complex conjugation, 9double layer potential, 202Fourier convolution, 48Fredholm, 3generalized Hankel, 50Laplace, 152Mellin, 49Mellin convolution, 49modified Lagrange, 131Muskhelishvili, 244projection, 24singular integral, 48Toeplitz, 172weighted singular integral, 50

operator function, 85

paired circulants, 76para-algebra, 38

factor, 39ideal, 39quotient, 39

para-group, 38

Poisson constant, 242polynomial

Lagrange, 65polynomial qualocation method, 100problem

biharmonic, 233biharmonic Dirichlet, 237generalized Riemann-Hilbert-

Poincare, 69Riemann-Hilbert, 123stability, 27

projection, 77initial, 30interpolation, 87orthogonal, 77range, 30

projection operator, 24

quadrature methods, 93qualocation method, 100, 170quotient algebra, 2quotient para-algebra, 39

range projection, 30real spectrum, 16reciprocally interchangeable systems,

46Riemann-Hilbert boundary problem,

123

self-adjoint element, 3separation condition, 30sequence

exact, 118Moore-Penrose stable, 32stable, 25weakly asymptotically Moore-

Penrose invertible, 33set M(A), 6set M(H), 10shift

Carleman, 10Sobolev space, 235Sohotsky-Plemely formulas, 243

306 Index

spaceHolder, 235Sobolev, 235weighted Lebesgue, 48, 54

spectral radiuscomplexified, 16

spectrum, 15complex, 16complexified, 16essential, 4real, 16

spline Galerkin Method, 91spline qualocation method, 104spline space, 87stable sequence, 25∗-algebra, 1Stechkin’s inequality, 49strictly real Banach algebra, 44strictly real C∗-algebra, 44, 58subalgebra

inverse closed, 17m-closed, 19

subpara-algebra, 39symbol

Gohberg-Krupnik, 84local, 83of sequence, 85

systemcovering, 41overlapping, 41

systemsreciprocally interchangeable, 46

tensor product, 81Toeplitz operator, 172transform

Fourier, 48Mellin, 49

unitalBanach algebra, 2para-algebra, 39

winding number, 173